Algebra II: Unit 13: Review

The area of a circle

(A = π r²) π(d²/4)

Binomial Coefficients

(n, r) or nCr or C(n, r) or C^nr nCr = n!/r!(n - r)! = n(n - 1)(n - 2)...(n - r + 1)/r! permutations of n objects taken r at a time; (n, r) = nPr/r!

Circle conic

(x - h)^2 + (y - k)^2 = r^2

The center-radius form of the circle equation

(x - h)^2 + (y - k)^2 = r^2 center; the point (h, k) radius; "r"

Ellipse conic

(x - h)^2/a^2 + (y - k)^2/b^2 = 1 (y - k)^2/a^2 + (x - h)^2/b^2 = 1

Hyperbola conic

(x - h)^2/a^2 - (y - k)^2/b^2 = 1 (y - k)^2/a^2 - (x - h)^2/b^2 = 1

the equation for a parabola

(y - mx - b)^2 / (m^2 +1) = (x - h)^2 + (y - k)^2 focus; (h, k) directrix; y = mx + b

Conjugates

* a + b ➡ a - b * x^2 - 3 ➡ x^2 + 3

Radians

1 radian = 180°/π π radians = 180° To go from radians to degrees: > multiply by 180, divide by π To go from degrees to radians: > multiply by π, divide by 180

Simplified form of radicals requires that:

1. No perfect n^th power factors of a radicand may exist in an n^th degree radical. 2. No fractions may exist in the radicand. 3. No radical may exist in the denominator of a fraction.

The process for determining the sine/cosine of any angleθ

1. Starting from (1,0)(1,0), move along the unit circle in the counterclockwise direction until the angle that is formed between your position, the origin, and the positive x-axis is equal to θ. 2. sin(θ) is equal to the y-coordinate of your point, and cos(θ) is equal to the x-coordinate.

Anti-log

10^n = x n: common log (base 10 log) x: inverse log 10^x; > the base for your antilog is always 10 > x is the number with which you are working example; 2.6452 > 2 = characteristic > 6452 = mantissa > 10^2.6452 = 441.7

The general equation for any conic section

Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 A,B,C,D,E and F are constants

combinations

C(n, r) = n!/r!(n - r)!

Radicals

I. (xy)^a = x^ay^a II. (x/y)^a = x^a/y^a III. a√x = x^1/a IV. √x = x^1/2 V. √xy = √x*√y VI. √x/y = √x/√y VII. √x^2 = x

The perimeter of a circle

P = 2πr πd

The Basic Trigonometric ratios

Sine = opp/hyp = a/c Cos = adj/hyp = b/c Tan = opp/adj = a/b

Imaginary number

a complex number that can be written as a real number multiplied by the imaginary unit i, which is defined by its property i2 = −1

Rational Powers

a^(1/n) = n√a a^(m/n) = (n√a)^m or n√(a^m)

The Discriminant

b^2 - 4ac of the quadratic formula b^2 - 4ac > 0; positive | 2 real roots b^2 - 4ac = 0; 1 real root b^2 - 4ac < 0; negative | no real roots

Natural Logarithms

base that is often used is e (Euler's Number) which is about 2.71828 generally written as ln x, loge x > base e is implicit InN = x ↔ N = e^x

Distance Formula

d = √(x2 - x1)^2 + (y2 - y1)^2

Arithmetic Sequence

example; -6, -1, 8, 15, 22 diff = 7 nth term; an = a1 + (n - 1) d

Geometric Sequence

example; 2, 4, 8, 16, 32 diff = *2 nth term; an = ar^n-1 > an = nth term of the sequence > a = first term of the sequence > r = common ratio > n = term number

The Reciprocal functions

general form of a reciprocal function; r(x) = a / (x - h) + k The vertical asymptote of r(x) is x = h The horizontal asymptote of r(x) is y = k csc = hyp/opp = c/a sec = hyp/adj = c/b cot = adj/opp = b/a

Common Logs

log x ↔ log10 x a logarithm is written without a base ; usually means that the base is really 10 logN = x ↔ N = 10^x

Series Notation

n ∑ an k = 1 n = last value an = formula for the terms k = index of summation 1 = first value of n

Permutations formula

n things taken r at a time: P(n, r) =n!/(n-r)! n things taken n at a time: P(n, n) = n! Circular permutation of n things taken r at a time: P^c(n, r)/r = n!/(n - r)!r Circular permutations of n things taken n at a time: P^c(n, n) = (n - 1)!

Factorials

n! = n*(n - 1)*(n - 2) n is a positive integer n! is the product of positive integers ≤ n

Rate

r = d/t > d = rt

The Unit Circle

raduis of 1 x^2 + y^2 = 1

conditional probability

represented by P(A/B) occurs when an event A occurs after event B has occurred

The Pythagorean trigonometric identity

sin2θ + cos2θ = 1 tan2θ + 1 = sec2θ 1 + cot2θ = csc2θ csc(x) = 1/sin(x) sec(x) = 1/cos(x) cot(x) = 1/tan(x) = cos(x)/sin(x) sin(x) = 1/csc(x) cos(x) = 1/sec(x) tan(x) = 1/cot(xx) = sin(x)/cos(x) sin(-t) = -sin(t) cos(-t) = cos(t) tan(-t) = -tan(t)

The Quadratic Formula

x = -b ± √b^2 - 4ac/2a

negative exponents

x^-n = 1/x^n x^n/x^m = x^n-m

Parabola conic

y = a(x - h)^2 + k x = a(y - k)^2 + h

Inverse Variation Equations

y = k/x k = xy

Direct Variation Equations

y = kx k = y/x

Logarithms

y = logb(x) ➡ b^y = x logb(a) = c ↔ b^c = a - b = base - c = exponent - a = argument logaN = x ↔ N = a^x

the "slope-intercept" form:

y=mx+b m = slope b = y-intercept